Pseudo Hermitian formulation of Black-Scholes equation

During the past few years there has been a great interest in studying problems of finance using various tools of physics [1]. In particular, different problems of finance have been studied from the point of view of quantum physics [2][3][4][5]. For example, various options have been modelled using quantum mechanical potentials [6], option pricing with stochastic volatility has been studied using the path integral technique [7]. Also, quantum mechanics has been used to analyze option pricing, stock market returns [8][9][10], Black-Scholes (BS) equation [11,12] etc. Supersymmetry formalism has been employed to obtain new solvable diffusion processes [13].

The BS equation (and its various generalizations) plays a dominant role in option pricing.

The solutions of the BS equation may be found by mapping it into a Schrödinger like equation. Then various quantities like the pricing kernel or the option price can be obtained using the solutions of the Schrödinger like equation. It may be pointed out that from the point of view of quantum mechanics the BS Hamiltonian is non Hermitian. On the other hand during the last decade non Hermitian quantum mechanics has been studied extensively.

A feature of such systems is that Schrödinger equation with many non Hermitian potentials admit real eigenvalues [14]. Subsequently it was shown that this unusual feature may be attributed to PT symmetry [14] or more generally to η-pseudo Hermiticity [15]. Here our objective is to show that the BS Hamiltonian and its various generalizations are η-pseudo Hermitian and we shall determine the explicit form of the metric operator η for each case.

We shall also show that the effective BS Hamiltonian together with its partner Hamiltonian [16] form a pseudo supersymmetric system.

The BS equation for option pricing with constant volatility is given by [2]

where C, S, σ and r denotes the price of the option, the stock price, the volatility of the stock price and the risk-free spot interest rate respectively. Now under the transformations [2] C(S, t) = e ǫt ψ(S) and S(x

the BS equation ( 1) becomes

where H BS is called the BS Hamiltonian. It is well known that the BS Hamiltonian in Eq.( 3) can be brought to the Schrödinger form [2,3]. To show this we use the similarity transformation

where

The Hamiltonian h BS in Eq.( 5) can be interpreted as a Schrödinger Hamiltonian of a particle of mass 1 σ 2 moving in a constant potential

It is important to note that the BS Hamiltonian in Eq.( 3) is non Hermitian while the Hamiltonian h BS in Eq.( 5) is Hermitian.

We shall now show that the BS Hamiltonian H BS is in fact η-pseudo Hermitian.

We recall that a Hamiltonian H is said to be η-pseudo Hermitian if

where η is a Hermitian operator. It has been shown that eigenvalues of a η-pseudo Hermitian

Hamiltonian are either completely real or occur in complex conjugate pairs [15]. In the context of financial modeling the BS equation usually has real eigenvalues and consequently it is of interest to examine the BS Hamiltonian from the point of view of pseudo Hermiticity.

Let us now define the metric operator η as

Then it can be verified that η = η † and

so that the BS Hamiltonian is η-pseudo Hermitian. Two important properties, namely, the completeness relation and the scalar product get modified for non Hermitian systems. In the present case they are given by

The above relations may be used to determine the pricing kernel. The pricing kernel p(x, τ, x ′ ) is defined as the conditional probability that the stock which has a value e x at time t will have a value e x ′ at time T = t + τ . The pricing kernel for the BS Hamiltonian is then given by

Then the option price is given by

where g(x) is the pay off function.

A. η-Pseudo Hermiticity of the Generalized BS Hamiltonian

Sometimes the BS Hamiltonian can be generalized by including a security dependent potential V (x). The resulting generalized Hamiltonian which satisfies the martingale condition is given by [2]

For an interpretation of the potential V (x) from the point of view of finance we refer the reader to ref [2]. Now, the generalized BS Hamiltonian in ( 12) is again non Hermitian. This can be seen from the fact that

The similarity transformation which transforms H into the Schrödinger form is given by

To show the η-pseudo Hermiticity of the generalized BS Hamiltonian H we define the metric operator η as in the last section i.e,

Then η = η † and after some calculations it can be shown that

so that the generalized BS Hamiltonian is η-pseudo Hermitian.

Path dependent options such as the Down-and-out barrier option, Out-and-out barrier option or the Double-knock-out barrier option can be analyzed by adding a potential term to the BS Hamiltonian (3) and the effective Hamiltonian is given by [2,8]

where H BS is given by (3). Clearly for a real potential V (x) we have

Since V (x) is real, it is clear that in this case the metric operator is given by ( 7) i.e,

In other words, the eff

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